Continuity

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Continuity

If a graph has no holes asymptotes, or breaks then the function is continuous. Or if you can draw the function without lifting your pencil then it is continuous. Below is a formal definition.

Definition of Continuity

A function is continuous at x = c if

$ \lim\limits_{x \to c}f(x) = f(c) $

Note that in order for a function to be continuous at a point, three things must be true:

The limit must exist at that point.
The function must be defined at that point, and
The limit and the function must have equal values at that point.

$Graph of a piecewise function with a vertical asymptote at x = -2, a hole at x = -1, a jump at x=0 from y=-1 to y=1 and filled in at y=5, and a jump at x=2 with the hole filled in at the right side.$

Notice that the function represented by the graph above is not continuous at

x = -2, x = -1, x = 0, and x = 2

Below is a list of function that are continuous.

Continuous Functions:

Polynomials
sin and cos
Rational Functions where the denominator is nonzero
Sums, Differences, and Products of continuous functions
Quotients of continuous functions where the denominator function is nonzero
Compositions of continuous functions

Examples:

The following are continuous:

y = x² + 3x - 4
y = x sin x
             1
y =
          1+ x²

Exercises:

Determine whether the following are continuous. If they are not continuous, at which points are they discontinuous?

          x - 1
y =
          x + 1
                x
y =
          x² + 3x - 4

y = {	2x + 3 for x < 1
	3x - 2 for x > 1

y = {	x² for x < 2
	5x - 6 for x > 2

For what value of k is the function continuous?

f(x) = {	3x² -5 for x < 1
	5x + k for x > 1

One Sided Limits

For a function with a break the limit does not exist, however it is still interesting to consider where the path is heading towards on the left side and where it heading on the right. For example if

                         |x|
          f(x) =
                          x

then for x negative

f(x) = -1

while for x positive,

f(x) = 1

We write

$ \lim\limits_{x \to 0^{-}} f(x) = -1 $

and

$ \lim\limits_{x \to 0^{+}} f(x) = 1 $

The Intermediate Value Theorem

Suppose a continuous function starts at the bottom left of the xy-plane and ends at the top right of the xy-plane. Now draw a horizontal line somewhere in the middle of the page. Can you draw a continuous function (that is without lifting the pencil from the paper) from the bottom left to the top right without crossing the line? The answer is certainly no. Try it! The intermediate value theorem formalizes this idea.

$x and y axes, start here is on the bottom left, just above the x-axis is a line that says "Do not cross this line" and End Here is on the top right.$

The Intermediate Value Theorem

If f is continuous on [a,b] and

f(a) < k < f(b)

then there exists at least one number c in the closed interval [a,b] for which

f(c) = k

In particular if f(a) and f(b) have different signs, then f has a root between a and b.

Example

Show that the curve defined by

y = 8x⁷ + x⁴ - 2x³ + x - 3

has a root between -1 and 1.

Solution

We apply the intermediate value theorem. The function

        f(x) = 8x⁷ + x⁴ - 2x³ + x - 3

is continuous between -1 and 1, since it is a polynomial. We have

        f(-1) = -8 + 1 + 2 - 1 - 3 = -9

and

        f(1) = 8 + 1 - 2 + 1 - 3 = 5

Since

        f(-1) < 0 < f(1)        Here k = 0

By the Intermediate Value Theorem, there is a c between -1 and 1 with

        f(c) = 0

Exercise:

Write psudocode to find a root of a function

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\( \lim\limits_{x \to c}f(x) = f(c) \)